$R$-matrix quantization of the elliptic Ruijsenaars–Schneider model

It is shown that the classical $L$-operator algebra of the elliptic Ruijsenaars–Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical $r$ and $\bar r$-matrices satisfying a closed system of equations. The corresponding quantum $R$ and $\overline R$-matrices are found as solutions to quantum analogs of these equations. We present the quantum $L$-operator algebra and show that the system of equations for $R$ and $\overline R$ arises as the compatibility condition for this algebra. It turns out that the $R$-matrix is twist-equivalent to the Felder elliptic $R^F$-matrix with $\overline R$ playing the role of the twist. The simplest representation of the quantum $L$-operator algebra corresponding to the elliptic Ruijsenaars–Schneider model is obtained. The connection of the quantum $L$- operator algebra to the fundamental relation $RLL=LLR$ with Belavin's elliptic $R$-matrix is established. Asa byproduct of our construction, we find a new $N$-parameter elliptic solution to the classical Yang–Baxter equation.